On the weak Borel chromatic number and cardinal invariants of the continuum

We prove that consistently, cov($\mathcal{M})<{\lambda }_{\mathbf{0}}<{\lambda }_{\mathbf{1}}<{\lambda }_{\mathbf{\infty }}<2^{{\mathrm{\aleph }}_0}$, where ${\lambda }_{\mathbf{0}}$ denotes the weak Borel chromatic number of the Kechris-Solecki-Todorčević graph ${\mathbb{G}}_0$, that is, the minimal cardinality of a ${\mathbb{G}}_0$-independent Borel covering of $2^{\omega }$, while ${\lambda }_{\mathbf{1}}$ and ${\lambda }_{\mathrm{\infty }}$ are the corresponding invariants of the graph ${\mathbb{G}}_1$ and the simple graph associated with the equivalence relation $E_0$.